Abstract: In this talk, we talk about well-posedness of the Cauchy problem to the forth-order nonlinear Schr odinger equations with derivative nonlinearities in the Sobolev space. The purpose of this talk is to improve the previous results obtained by several Mathematicians, that is, to treat more general nonlinearity and to prove well-posedness of the problem in lower order Sobolev space. Our proof of the well-posedness result is based on the contraction argument on a suitable function space, via the Strichartz estimates, Kato-type smoothing estimates, Kenig-Ruiz estimates, Maximal function estimates, a linear estimate for inhomogeneous term, the bilinear Strichartz type estimate and the Littlewood-Paley theory. This talk is based on a joint work with Hiroyuki Hirayama and Tomoyuki Tanaka.